# Thread: Find two positive divergent sequences such that..

1. ## Find two positive divergent sequences such that..

Hello
Find $\displaystyle a_n$ and $\displaystyle b_n$ such that:
{$\displaystyle an$} , {$\displaystyle b_n$} are divergent sequences
$\displaystyle a_n,b_n > 0$ for $\displaystyle n\geq1$
and { $\displaystyle a_n+b_n$ } is a convergent sequences.

The problem here is both must positive.
If not, $\displaystyle a_n=n$ and $\displaystyle b_n=-n$ is a nice counter-example.

But they must be positive.

2. Originally Posted by General
Hello
Find $\displaystyle a_n$ and $\displaystyle b_n$ such that:
{$\displaystyle an$} , {$\displaystyle b_n$} are divergent sequences
$\displaystyle a_n,b_n > 0$ for $\displaystyle n\geq1$
and { $\displaystyle a_n+b_n$ } is a convergent sequences.

The problem here is both must positive.
If not, $\displaystyle a_n=n$ and $\displaystyle b_n=-n$ is a nice counter-example.

But they must be positive.

Here's a tip, just because a sequence is divergent it doesn't mean it goes to infinity.

Bigger hint;
Spoiler:
think sine and cosine.

3. Originally Posted by pomp
Here's a tip, just because a sequence is divergent it doesn't mean it goes to infinity.

Bigger hint;
Spoiler:
think sine and cosine.
But sine and cosine are not positive for all $\displaystyle n\geq1$

4. Originally Posted by General
But sine and cosine are not positive for all $\displaystyle n\geq1$
Well then do something to them that will ensure they are...

5. You mean $\displaystyle sin^2n$ and $\displaystyle cos^2n$ ?
What the limit of them ?
What the limit of thier sum ?
as $\displaystyle n \rightarrow \infty$

6. Originally Posted by General
You mean $\displaystyle sin^2n$ and $\displaystyle cos^2n$ ?
What the limit of them ?
What the limit of thier sum ?
as $\displaystyle n \rightarrow \infty$
"What the limit of them?"

Neither converge to a limit, they oscillate in (0,1)

"What is the limit of thier sum? "

What is $\displaystyle \sin^2 (x) + \cos^2(x)$ ?